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Numerical methods

Department of Mathematics, University of Oslo

My (now finished) PhD student Miroslav Kuchta has been looking at numerical methods to solve saddle point systems arising from trace constraints coupling 2D and 1D domains, or 3D and 1D domains Kuchta et al. (2016)Kuchta et al. (2018)Kuchta et al. (2015).

We have also been studying the singular Neumann problem of linear elasticity Kuchta et al. (2018). Four different formulations of the problem have been analyzed and mesh independent preconditioners established for the resulting linear systems within the framework of operator preconditioning. We have proposed a preconditioner for the (singular) mixed formulation of linear elasticity, that is robust with respect to the material parameters. Using an orthonormal basis of the space of rigid motions, discrete projection operators have been derived and employed in a modification to the conjugate gradients method to ensure optimal error convergence of the solution.

With colleagues at the Extreme Computing Research Center (ECRC), King Abdullah University of Science and Technology (KAUST), we have been using spectralDNS to investigate time integration of Fourier pseudospectral Direct Numerical Simulations Ketcheson et al. (2020). We investigate the use of higher‐order Runge‐Kutta pairs and automatic step size control based on local error estimation. We find that the fifth‐order accurate Runge‐Kutta pair of Bogacki and Shampine gives much greater accuracy at a significantly reduced computational cost.

In the SISC paper Mortensen (2023) I describe a very efficient strictly banded method for solving differential equations with polynomial coefficients. This method can be understood as a Petrov-Galerkin description of the integration preconditioner method by Coutsias et al. (1995). The method has been implemented in Shenfun.

Also in SISC, the paper Mortensen (2024) describes the fastest known Legendre to Chebyshev (and vice versa) transform, based on the fast multipole method of Alpert and Rokhlin Alpert & Rokhlin (1991). This method is implemented in the github repository SISC-Legendre-to-Chebyshev.

References

References
  1. Kuchta, M., Nordaas, M., Verschaeve, J. C. G., Mortensen, M., & Mardal, K.-A. (2016). Preconditioners for Saddle Point Systems With Trace Constraints Coupling 2D and 1D Domains. SIAM Journal on Scientific Computing, 38(6), B962–B987. 10.1137/15M1052822
  2. Kuchta, M., Mardal, K.-A., & Mortensen, M. (2018). Preconditioning Trace Coupled 3d-1d Systems Using Fractional Laplacian. Numer. Methods Partial Differential Equations. https://arxiv.org/abs/1612.03574
  3. Kuchta, M., Mardal, K. A., & Mortensen, M. (2015). Characterization of the Space of Rigid Motions in Arbitrary Domains. In Bjørn Helge Skallerud and Helge Ingolf Andersson (Ed.), MekIT’15 - Eight National Conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE), 259–274.
  4. Kuchta, M., Mardal, K.-A., & Mortensen, M. (2018). On the Singular Neumann Problem in Linear Elasticity. Numerical Linear Algebra with Applications. https://arxiv.org/abs/1609.09425
  5. Ketcheson, D. I., Mortensen, M., Parsani, M., & Schilling, N. (2020). More efficient time integration for Fourier pseudospectral DNS of incompressible turbulence. International Journal for Numerical Methods in Fluids, 92(2), 79–93. 10.1002/fld.4773
  6. Mortensen, M. (2023). A Generic and Strictly Banded Spectral Petrov–Galerkin Method for Differential Equations with Polynomial Coefficients. SIAM Journal on Scientific Computing, 45(1), A123–A146. 10.1137/22M1492842
  7. Mortensen, M. (2024). A Faster Multipole Legendre–Chebyshev Transform. SIAM Journal on Scientific Computing, 46(6), A3803–A3826. 10.1137/24M1659352
  8. Alpert, B. K., & Rokhlin, V. (1991). A fast algorithm for the evaluation of Legendre expansions. SIAM Journal on Scientific and Statistical Computing, 12(1), 158–179.
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